Takeoff vs. Stagnation in Endogenous Recombinant
Growth Models
Fabio Privileggi
∗August 27, 2013
Abstract
This paper concludes the study of transition paths in the continuous-time recombi-nant endogenous growth model by providing numerical methods to estimate the threshold initial value of capital (a Skiba-type point) above which the economy takes off toward sus-tained growth in the long run, while it is doomed to stagnation otherwise. The model is based on the setting first introduced by Tsur and Zemel and then further specified by Privileggi, in which knowledge evolves according to the Weitzman recombinant process. We pursue a direct approach based on the comparison of welfare estimations along optimal consumption trajectories either diverging to sustained growth or converging to a steady state. To this purpose, we develop and test three algorithms capable of numerically sim-ulating the initial Skiba-value of capital, each corresponding to initial stock of knowledge values belonging to three different ranges, thus covering all possible scenarios.
JEL Classification Numbers: C61, C62, C63, C68, O31, O41.
Key words: Knowledge Production, Endogenous Recombinant Growth, Transition Dynamics, Turnpike, Skiba Point, Hamilton-Jacobi-Bellman equation.
1
Introduction
This paper provides a further contribution to the study of the two-sector continuous-time en-dogenous growth model introduced by Tsur and Zemel [24] in which knowledge evolves accord-ing to the Weitzman [25] recombinant process. Given any feasible initial stock of knowledge, we provide three numeric algorithms capable of approximating the corresponding critical initial capital value above which the economy “takes off” toward an asymptotic balanced growth path (ABGP), while it is led toward a steady point – i.e., to stagnation in the long run – whenever the initial capital lies below such threshold. To this purpose we elaborate on the functional forms introduced by Privileggi [19], which are suitable to ‘detrend’ the model and thus obtain a closed form for the ODE defining the optimal policy that, in turn, can be approximated with a sufficient degree of accuracy by means of a projection method discussed in [20].
Weitzman [25] departs from the mainstream endogenous growth literature1 flourished after
the original works of either Romer [21], [22] – based on technology spillovers/externalities –
∗Dept. of Economics and Statistics “Cognetti de Martiis”, Universit`a di Torino, Lungo Dora Siena 100 A,
10153 Torino (Italy). Phone: +39-011-6702635; fax: +39-011-6703895; e-mail fabio.privileggi@unito.it
1For recent comprehensive surveys see [4], [1] and, more oriented toward the creative-destruction
or Grossman and Helpman [8] and Aghion and Howitt [2] – building on the Shumpeterian tradition of the creative-destruction process involved in innovation activities – by focussing on two peculiar elements that drive knowledge generation: the process of creation of new ideas and the resources needed to turn these ideas into “productive” knowledge. The evolution of ideas is assumed to follow a recombinant mechanism: existing ideas are combined (matched) to generate new ideas. The number of possible matchings is a combinatorial function of the number of existing ideas that spontaneously would give rise to an unrealistic over-exponential growth. As a matter of fact, such explosive dynamic is contained by the fact that turning potentially fruitful ideas into useful knowledge requires physical resources, whose optimal allocation by a social planner has been first analyzed by Tsur and Zemel [24] in a continuous-time setting.
We consider a specification of the model in [24] where the probability of a successful matching among existing seed ideas has a hyperbolic form, a composite final good is produced in a competitive sector by means of a Cobb-Douglas function using the stock of knowledge and physical capital as input factors, and the representative household has a CIES utility function. A social planner efficiently maximizes the discounted utility of a representative consumer over an infinite time horizon by directly financing new knowledge production through a tax levied on the households; at each instant new knowledge is produced by an independent R&D sector under the supervision of the social planner. Hence, we are pursuing a first-best, social planner-type equilibrium approach, setting aside all issues regarding incentives to innovate, spillovers, externalities, etc., and the related scale effects involved by knowledge production.2
This hyperbolic-Cobb-Douglas-CIES specification of the model allows for a closed-form ODE defining the optimal transition dynamics (see [19]) along a characteristic curve in the knowledge-capital state space that will be labeled as (transitory) turnpike when the conditions for sustained long-run growth provided by [24] are met. The solution of such ODE is numerically approximated through an appropriate projection method (see [20]) and can be used to compute, by means of a finite-difference, Runge-Kutta method, the optimal time-path trajectories of the stock of knowledge, capital, output and consumption, as well as their transition growth rates, along the turnpike. However, whenever the initial capital is different than its unique value on the turnpike, different types of transition paths appear; they can either reach a point on the turnpike in a finite time period and then continue along the turnpike itself toward sustained growth, or can converge to a steady state which is a point on another characteristic curve in the knowledge-capital state space that will be called the stagnation line.
The aim of this paper is to thoroughly investigate the latter type of (initial) transitions. Tsur and Zemel [24] showed that, for each given initial stock of knowledge, there corresponds a unique critical value for the initial capital such that for any value above this threshold the economy will first follow a path toward the turnpike and then, along a path evolving along the turnpike itself, toward sustained growth along a ABGP. Conversely, when the initial capital is below such threshold, the process generating new knowledge does not take off and the economy eventually dies in stagnation by converging asymptotically to steady values for both knowledge and capital on the stagnation line. The properties of this threshold value are akin to those first discussed by Skiba [23]; hence we shall refer to this point as the Skiba-point.
We first develop a numerical method (Algorithm 1) that computes the Skiba-point on the turnpike, labeled as Am, kskm, by equating the welfare when sustained growth is triggered
with the welfare associated to a path leading toward stagnation, starting from the same point Am, kmsk. Next, we consider initial values of the stock of knowledge, A0, which lie on the left of
Am and build a more complex Bisection method (Algorithm 2), again with the goal of matching
the welfare when taking off toward the ABGP with the welfare of the economy converging to
a steady state, to find the Skiba-point when the economy starts on an initial capital, k0, lying
above the turnpike value corresponding to A0. Finally, we focus on initial values A0 lying to
the right of Am; in this case, to estimate the Skiba-point we propose another Bisection method
(Algorithm 3) with the aim of equating the welfare generated by the trajectory that starts from an initial capital, k0, below the turnpike, climbs up toward the turnpike, reaches it in a finite
time period, and then keeps following it thereafter toward the ABGP, with that produced by the trajectory converging to a steady state starting from the same initial point (A0, k0).
All optimal trajectories are estimated through a mix of projection methods and Runge-Kutta type algorithms. First a projection method – based either on OLS or on Orthogonal Collocation and with a residual function defined by means of Chebyshev polynomials (see, e.g., Chapter 11 in [16], Chapter 6 in [12], or Paragraph 5.5.2 in [18]) – is applied to the ODE defining the optimal policy. The approximation thus obtained can then be used in a Runge-Kutta method to generate all transition time-path trajectories. Welfare estimates along the turnpike or toward stagnation are performed through direct computation of the value function by means of the Hamilton-Jacobi-Bellman equation in which the derivative of the value function is calculated through the celebrated Benveniste and Scheinkman [5] result as the derivative of the instantaneous utility at the initial optimal consumption value. While this technique is immediately available for the dynamics converging to a steady state because in this case the model boils down to a standard concave Ramsey-type model, for the dynamics along the turnpike, eventually leading to steady growth, we must rely on the Hamilton-Jacobi verification principle to establish that the Hamilton-Jacobi-Bellman equation actually delivers the true value function (Proposition 4), as in this case the model turns out to be not concave in early-times. Along initial trajectories outside the turnpike eventually reaching it after a finite time period, welfare is estimated trough Gauss-Legendre quadrature routines, themselves built on the simulations of the consumption time-path trajectories previously calculated.
All simulations produce a rich variety of early-transition dynamic patterns, which are inter-esting per se. Our main findings are summarized in a plot in the knowledge-capital space re-porting a number of estimated Skiba-points (Figure 8); the figure suggests that all Skiba-points lie on a decreasing curve plunging to zero as the initial knowledge approaches the intersection point between the turnpike and the stagnation line. However, the performances of Algorithm 3 rapidly degenerate as the initial knowledge level approaches this intersection point.
Because the set of Skiba-points turns out to be a curve, our results contribute to the literature, started by Haunschmied et al. [11], focussed on numerically computing the DNS-curve (so named in honour of the pioneering works of Skiba [23] and Dechert and Nishimura [7] who first introduced the notion of Skiba-point) separating the basins of attraction of different locally stable steady states (or cycling orbits) in continuous-time economic models.3
Section 2 reports some well known preliminary results that will be used throughout the paper. Section 3 recalls the main facts related to endogenous recombinant growth according to Weitzman [25] and Tsur and Zemel [24], and introduces the specification of Privileggi [19], [20]. Section 4 focusses on the Skiba-point Am, kmsk on the turnpike, characterizing the
opti-mal dynamics along the turnpike and toward stagnation, and describes our welfare estimation techniques for this case. Sections 5 and 6 characterize the early transition dynamics outside the turnpike, leading to the elaboration of two algorithms for Skiba-point estimations above and below the turnpike respectively. In Section 7 all the algorithms are then used to approximate all types of Skiba-points for a specific example. Finally, Section 8 concludes, while the Appendix contains the proof of our main theoretical result (Proposition 4).
2
Preliminaries
Here a few well known results that will be used throughout the paper for welfare evaluation purposes are reported without proof. Given a technology set T ⊆ R2n, consider the standard
continuous-time problem
V (x0) = sup
Z ∞ 0
e−ρtU [x (t) , ˙x (t)] dt (1) subject to [x (t) , ˙x (t)] ∈ T for all t and x (0) = x0,
where ρ > 0 is the discount rate and U (·, ·) is the instantaneous felicity. Lemma 1 (Principle of optimality) Suppose that (x∗
(t; x0) , ˙x∗(t; x0)) is a solution of (1)
originating at x (0) = x0. Then, for all t0 ≥ 0
V (x0) =
Z t0 0
e−ρtU [x∗
(t; x0) , ˙x∗(t; x0)] dt + e−ρt0V [x∗(t0; x0)] . (2)
We denote by T (x) the x-section of the set T , i.e., T (x) = {(x, ˙x) ∈ R2n : (x, ˙x) ∈ T }.
Theorem 1 (Hamilton-Jacobi verification principle) Assume that:
(i) w : Rn→ R is of class C1 and satisfies the Hamilton-Jacobi-Bellman equation, i.e.,
ρw (x) = max
˙x∈T (x)[U (x, ˙x) + ∇w (x) · ˙x] ; (3)
(ii) for every initial condition x0 there is a feasible ˙x∗(t; x0) such that the max is attained in
(3), i.e.,
ρw [x∗
(t; x0)] = U [x∗(t; x0) , ˙x∗(t; x0)] + ∇w [x∗(t0; x0)] · ˙x∗(t; x0) (4)
for all t ≥ 0, a.e.;
(iii) limt→∞e−ρtw [x (t; x0)] = 0 for every feasible path x (t; x0).
Then w (x) is the value function of (1), i.e., V (x) = w (x), and (x∗(t; x
0) , ˙x∗(t; x0)) is a
solution of (1).
Theorem 2 (Benveniste and Scheinkman) Assume that: (i) T is convex and int (T ) 6= ∅;
(ii) U : T → R is continuously differentiable on int (T ) and concave; (iii) an optimal solution x∗
(t; x0) from x0 (not necessarily unique) exists and V (x) in (1) is
defined in some neighborhood of x0;
(iv) the optimal solution is interior in the following sense: there exist h > 0, ε > 0 and M > 0 such that for t ∈ [0, h], kx∗
(t; x0) , ˙x∗(t; x0)k ≤ M and if (z, z′) ∈ R2n satisfies
k(x∗
(t; x0) , ˙x∗(t; x0)) − (z, z′)k ≤ ε for some t ∈ [0, h], then (z, z′) ∈ T ;
(v) ˙x∗
Then V (x) in (1) is of class C1 at x 0 and
∇V (x0) = −∇˙xUx0, ˙x∗ 0+; x0 , (5)
where ∇˙xU (·, ·) denotes the vector of partial derivatives of U with respect to its second argument.
A proof of Theorem 2 can be found in [5]. The following corollary provides a converse result of Theorem 1 under the value function differentiability provided by Theorem 2.
Corollary 1 Under the same assumptions of Theorem 2 the value function V (x) in (1) satisfies the Hamilton-Jacobi-Bellman equation (3), and the maximum is attained at ˙x = ˙x∗
(0+; x
0), i.e.,
ρV (x0) = U x0, ˙x∗ 0+; x0 + ∇V (x0) · ˙x∗ 0+; x0 . (6)
3
The Model
In continuous-time the flow of successful new ideas accruing the existing stock of knowledge is given by
˙
A (t) = H (t) π [J (t) /H (t)] , (7) where A (t) is the stock of knowledge at time t (measured as the total number of fruitful ideas),
˙
A (t) denotes its time-derivative, H (t) is the number of yet unprocessed (seed ) ideas at instant t which are combined together in order to obtain new hybrid seed ideas of which only a fraction turns out to be successful, according to a probability function π (·) that itself depends on the ratio between a measure of the physical resources devoted to the R&D recombinant process, J (t), and the available seeds, H (t), at instant t (see [24], [19] and [20]).
A. 1 The success probability function is independent of time and is given by4
π (x) = βx/ (βx + 1) , β > 0. (8) Parameter β is a measure of the ‘degree of efficiency’ of the recombinant process: the larger β the higher the probability of obtaining a new successful idea out of each seeds matching. The continuous-time setting implies that ˙A (t) has the same value both while looking forward to the new output – equation (7) – and while looking backward, i.e., to the formation of seed ideas, which is given by
H (t) = C′
m[A (t)] ˙A (t) , (9)
where Cm(A) = A!/ [m! (A − m)!] is of the number of different combinations of m seed ideas as
a function of the stock A and C′
m(A) denotes its derivative. (9) is the continuous-time version
of (26) on p. 345 in [25]. We assume that only pairs of seed ideas are combined together: m = 2. Hence, Cm(A) = C2(A) = A (A − 1) /2 and Cm′ (A) = C
′
2(A) = A − 1/2, so that (9)
boils down to
H (t) = [A (t) − 1/2] ˙A (t) . (10) Within this approach both the seed production in (10) and the production of new ideas in (7) are referred (as a limit) to the same time instant, so that (10) can be substituted into (7) to yield the following law of motion for the stock of knowledge:
˙
A (t) = J (t) /ϕ [A (t)] , (11)
4π : R
+ → [0, 1) in (8) satisfies Weitzman’s assumptions (p. 345 in [25]): π′ > 0, π′′ < 0, π (0) = 0 and
where, under Assumption A.1 and for m = 2, ϕ (A) = C′
2(A) π−1[1/C ′
2(A)] = (1/β) [1 + 2/ (2A − 3)] , (12)
is the expected unit cost of knowledge production, which is defined for5 A > 3/2, is decreasing
in A, and limA→∞ϕ (A) = 1/π′(0) = 1/β > 0.
The social planner chooses the optimal amount J to be employed in production of new knowledge according to (11) in order to maximize the discounted utility of a representative consumer over an infinite time horizon. J is levied as a tax on the representative consumer, and the new “productive” knowledge obtained is immediately and freely passed to the output producing firms. We assume that labor is constant and normalized to one: L ≡ 1.
A. 2 Output is produced according to a Cobb-Douglas technology:
y (t) = θ [k (t)]α[A (t)]1−α, θ > 0, 0 < α < 1, (13) depending on aggregate capital, k (t), and knowledge-augmented labor, A (t) L (t), for L (t) ≡ 1. Output producing firms maximize instantaneous profit by renting capital k and hiring labor L from the households, taking as given the capital rental rate, r, the labor wage and the stock of knowledge, A. As these firms operate in a competitive market, it follows from A.2 that:
θα [k (t) /A (t)]α−1 = r (t) . (14) We slightly depart from [19] and [20] by setting an upper bound on investment in R&D activities: J (t) ≤ y (t) for all t ≥ 0.6 Hence, capital evolves through time according to
˙k (t) = y (t) − J (t) − c (t) , (15) where it is assumed that capital does not depreciate.
A. 3 All households enjoy an instantaneous CIES utility,
u (c) = c1−σ − 1 / (1 − σ) , σ ≥ 1, (16) and have a common discount rate, ρ > 0.
Thus, the welfare maximization problem faced by the social planner is V (A0, k0) = max [c(t),J(t)]∞ t=0 Z ∞ 0 e−ρt[c (t)] 1−σ − 1 1 − σ dt (17) subject to the dynamic constraints (11) and (15), with the additional constraints J (t) ≤ y (t), c (t) ≤ k (t) + y (t), and usual non-negativity constraints, given the initial stock of physical capital, k0 > 0, and knowledge, A0 > 3/2. Suppressing the time argument, the current-value
Hamiltonian associated to (17) is
H (A, k, J, c, λ, δ) = c1−σ− 1 / (1 − σ) + λ θkαA1−α
− J − c + δJ/ϕ (A) , (18)
5Note that the RHS in (12) is negative for 1/2 < A < 3/2, while for A ≤ 1 the interpretation of the
Weitzman’s process based on the combination of (more than 1) ideas becomes meaningless.
6As in the original framework of [25] and [24], the social planner cannot spend more than the aggregate
where λ and δ are the costate variables associated with k and A respectively and ϕ (A) is defined by (12). Necessary conditions are:7
u′ (c) = λ J = 0 if δ/ϕ (A) < λ ˜ J if δ/ϕ (A) = λ θkαA1−α if δ/ϕ (A) > λ (19) ˙λ = ρλ − λθ (k/A)α−1 ˙δ = ρδ − λθ (1 − α) (k/A)α + δJϕ′ (A) / [ϕ (A)]2 lim t→∞H (t) e −ρt = 0,
where ˜J in (19) is defined by (23) below.
Remark 1 While the costates λ and δ are continuous functions of time,8 conditions (19)
im-ply a discontinuous optimal R&D financing (a ‘ bang-bang’ solution) due to linearity of the Hamiltonian (18) in the variable J. On the other hand, the necessary condition c−σ = λ and
continuity of λ in time implies that the optimal trajectory of consumption must be a continuous function of time.
The solution of (17) in this regulated economy is described by means of the following three characteristic curves in the space (A, k).
1. The locus on which the marginal product of capital equals that of knowledge per unit cost, which, under Assumptions A.1 and A.2, using (12) can be rewritten as a function of the only variable A:
˜
k (A) = [α/ (1 − α)] ϕ (A) A = {α/ [β (1 − α)]} [1 + 2/ (2A − 3)] A. (20) We call ˜k (A) in (20) the (transitory) turnpike.
2. The function ˜k (A) in (20) for large A becomes affine, defining the curve ˜
k∞(A) = {α/ [β (1 − α)]} (A + 1) . (21)
We call ˜k∞(A) in (21) the asymptotic turnpike. ˜k (A) lies above ˜k∞(A), that is, ˜k (A) >
˜
k∞(A) for all A < ∞, and approaches ˜k∞(A) as A → ∞.
3. Finally, on the locus θα (k/A)α−1 = ρ the marginal product of capital equals the individual discount rate, which, by (14), implies r = ρ. It can be written as a linear function of A:
ˆ
k (A) = (θα/ρ)1/(1−α)A. (22) We call ˆk (A) in (22) the stagnation line, as it defines the set of all possible steady pairs (k, A) on which the economy might eventually end up in stagnation.
7See conditions (27) – (31) in [24] or conditions (15) – (19) in [19].
Differentiating ˜k (A) in (20) with respect to time and substituting into the dynamic con-straints (11) and (15) yields
˜
J = (˜y − ˜c) ϕ (A) /h˜k′
(A) + ϕ (A)i, (23) where ˜y = θh˜k (A)iαA1−α, and ˜c denotes optimal consumption when the economy is
con-strained to grow along the turnpike ˜k (A). Condition (23) relates the optimal investment in R&D, ˜J, as a function of the other control variable, ˜c, along the transitory turnpike; that is, in view of (19), when δ/ϕ (A) = λ holds.
Proposition 1
i) A necessary condition for the economy to sustain long-run growth is
ρ < θα [β (1 − α) /α]1−α, (24) where the RHS, θα [β (1 − α) /α]1−α, defines the long-run capital rental rate.
ii) Under (24), for any given initial knowledge stock A0 > 3/2 there is a unique corresponding
threshold capital stock ksk(A
0) ≥ 0, to which we refer as the Skiba-point, such that
whenever k0 ≥ ksk(A0) the economy first reaches the turnpike ˜k (A) in a finite time period,
and then continues to grow along it until the asymptotic turnpike ˜k∞(A) is approached
as A → ∞. Along ˜k∞(A) the economy follows a ABGP characterized by the following
common constant growth rate of output, knowledge, capital and consumption:
γ =θα [β (1 − α) /α]1−α− ρ /σ. (25) Moreover, there exist an instant t0 ≥ 0 such that J (t) > 0 for all t > t0, while, as
t → ∞, J (t) < y (t) holds and the income shares devoted to investments in knowledge and capital are constant and given by s∞ = (1 − α) γ/θα [β (1 − α) /α]
1−α
and sk
∞ =
αγ/θα [β (1 − α) /α]1−α respectively.
iii) Conversely, whenever either (24) fails, i.e., if ρ ≥ θα [β (1 − α) /α]1−α, or k0 < ksk(A0)
after a finite instant ts ≥ 0 the optimal investment in R&D activities becomes zero and
the stock of knowledge remains constant: J (t) ≡ 0 and A (t) ≡ A (ts) for all t > ts. In
this scenario – perhaps after a time interval of “full investment” in R&D, during which J (t) = y (t), if δ (t) /ϕ [A (t)] > λ (t) holds in (19) for 0 ≤ t ≤ ts – eventually optimal
capital and consumption follow the usual Ramsey-type saddle-stable time-path trajectory monotonically converging to a steady state on the stagnation line ˆk (A) defined in (22), with steady value for the physical capital equal to ˆk [A (ts)], corresponding to zero growth.
For a proof see [24]. Figure 1 shows all three characteristic curves for the parameters’ values considered in Section 7 satisfying condition (24).
4
The Unique Skiba-Point on the Turnpike
Proposition 1 (ii) implies that, under (24) and if k0 ≥ ksk(A0), the turnpike ˜k (A) is ‘trapping’,
i.e., the economy keeps growing along it after it is entered so to reach the asymptotic turnpike ˜
k∞(A) for t → ∞ and follow the ABGP thereafter. In order to estimate the Skiba-point
ksk(A
A k Aℓ Aˆ ˆ k kℓ k (A)˜ ˆk (A) ˜ k∞(A)
Figure 1: the transitory turnpike ˜k (A) (in black), the stagnation line ˆk (A) (in dark grey) and the asymptotic turnpike ˜k∞(A) (in light grey) of our economy for the parameters’ values used in Section
7; ˆA, ˆkis the intersection point between the transitory turnpike and the stagnation line.
1. the path driving the system toward the turnpike starting from outside it, and 2. the path characterizing the optimal path along ˜k (A) after it has been entered.
Special attention will be devoted on the former, as the latter has been already thoroughly analyzed in [19] and [20].
First, we need to narrow the range of our analysis according to the following preliminary result. From (21) and (22) it is immediately seen that the growth condition (24) states that the slope of the asymptotic turnpike ˜k∞(A) must be less than the slope of the stagnation line
ˆ
k (A); because the transitory turnpike ˜k (A) lies above ˜k∞(A) for all finite A and (20) implies
limA→3/2+˜k (A) = +∞, there is a value ˆA at which the turnpike ˜k (A) intersects the stagnation
line ˆk (A), that is, such that ˜k ˆA= ˆk ˆA. Such value is unique and is obtained by coupling (20) and (22):
ˆ
A = α/hβ (1 − α) (θα/ρ)1−α1 − α
i
+ 3/2, (26)
which is well defined whenever the necessary condition for growth (24) is satisfied. Figure 1 illustrates this property for the parameters’ values considered in Section 7.
Proposition 2 Under growth condition (24) for all initial stock of knowledge levels A0 ≥ ˆA
the economy is bound to sustain growth in the long run independently of the initial stock of capital k0 > 0; that is, whenever A0 ≥ ˆA, ksk(A0) = 0.
Proof. It is an immediate consequence of Property 4b on p. 3472 in [24], stating that a steady state cannot lie above the turnpike ˜k (A).
Hence, given ˆA defined in (26), we confine our attention to levels 3/2 < A0 < ˆA for the
physical capital corresponding to A0, ˜k (A0), which lies on the turnpike. Clearly, given A0,
the Skiba-point ksk(A
0) can either lie above or below the number ˜k (A0). However, there is
also the very peculiar case in which ksk(A
0) = ˜k (A0). As the two cases ksk(A0) > ˜k (A0) and
ksk(A
0) < ˜k (A0) exhibit quite diverse types of transition dynamics, the scenario ksk(A0) =
˜
k (A0) determines a boundary value separating these two cases. We start by investigating this
boundary regime; specifically, we look for the initial knowledge stock level A0 that, when the
economy starts with an initial capital endowment k0 = ˜k (A0), the welfare generated by growing
along ˜k (A) and then converging toward a ABGP along ˜k∞(A) equals the welfare obtained by
converging toward the steady state A0, ˆk (A0)
on the stagnation line along a monotonic Ramsey-type saddle-stable time-path trajectory.
Proposition 3 If a knowledge value 3/2 < ¯A ≤ ˆA exists such that the optimal dynamics converge to a steady state on the stagnation line ˆk (A) when the economy starts from ¯A, ˜k ¯A on the turnpike, then for all 3/2 < A0 ≤ ¯A it is optimal to converge to a steady state on the
stagnation line when the economy starts from A0, ˜k (A0)
on the turnpike.
Proof. It follows immediately from Property 6 on p. 3473 in [24], establishing that the singular policy (23) along the turnpike is trapping.
Proposition 3 implies that, if it exists, there is a unique minimal knowledge level 3/2 < Am ≤ ˆA such that when the economy initiates from
Am, ˜k (Am)
on the turnpike, it is bound to proceed along the turnpike toward sustained long-run growth. Indeed, such Am
value corresponds to the unique Skiba-point lying on the turnpike, i.e., the unique knowledge level satisfying ksk(A
m) = ˜k (Am) we are looking for. Hence, Am is the initial level of the stock
of knowledge that equates the welfare obtained by investing ˜J (t) > 0 as in (23) for all t ≥ 0 and thus following the optimal trajectory A (t) , ˜k [A (t)] on the turnpike starting from the point Am, ˜k (Am)
, with the welfare yield, according to δ/ϕ (A) < λ in (19), by a zero-R&D investment policy, J (t) ≡ 0 for all t ≥ 0, keeping constant the stock of knowledge at its initial level Am and leading the economy toward the steady state
Am, ˆk (Am)
on the stagnation line. To estimate Am we thus evaluate the welfare delivered both by trajectories evolving along
the turnpike ˜k (A) following the policy (23) and trajectories evolving toward the steady state
A0, ˆk (A0)
on the stagnation line through a zero-R&D investment policy – as J (t) ≡ 0 implies A (t) ≡ A0 for all t ≥ 0.
4.1
Optimal Dynamics Along the Turnpike
Optimal trajectories along the turnpike are solutions of the following social planner problem in the only two variables A (state) and c (control), and one dynamic constraint:
˜ V (A0) = max [c] Z ∞ 0 e−ρtc1−σ − 1 1 − σ dt (27) subject to ( ˙ A =nθh˜k (A)iαA1−α− co/h˜k′ (A) + ϕ (A)i A (0) = A0, (28) where the time argument has been dropped for simplicity, ˜k (A) is defined in (20), ˜k′(A) =
for problem (27) yield the following system of ODEs defining the optimal dynamics for A and c along the turnpike:
˙
A =nθAh˜k (A) /Aiα− co/h˜k′
(A) + ϕ (A)i ˙c = c θαh˜k (A) /Aiα−1− ρ /σ. (29)
To study the phase diagram associated to (29), using (20) and (12) Privileggi [19] introduces the ratio variables
µ = ˜k (A) /A = [α/ (1 − α)] ϕ (A) = {α/ [β (1 − α)]} [1 + 2/ (2A − 3)] , (30)
χ = c/A, (31)
which transform (29) into the following system of ODEs: (
˙µ = [1 − 2β (1 − α) µ/Q (µ)] (θµα− χ)
˙χ = [(θαµα−1− ρ) /σ − 2αβ (1 − α) (θµα− χ) /Q (µ)] χ, (32)
where
Q (µ) = −3β2(1 − α)2µ2+ 2β (1 − α) (1 + 2α) µ − α2. (33) Unlike system (29), whose variables A and c diverge in the long-run, µ and χ solving (32) can converge to the steady state (µ∗
, χ∗
) whose coordinates are defined by µ∗
= α/ [β (1 − α)] and χ∗
= θ {α/ [β (1 − α)]}α(1 − 1/σ) + ρ/ [βσ (1 − α)] , (34) where µ∗
is to the constant long-run capital/knowledge ratio along the asymptotic turnpike ˜
k∞(A) [µ ∗
is the slope of ˜k∞(A) in (21)] and χ ∗
is the long-run consumption/knowledge ratio. (µ∗, χ∗) is saddle-path stable, with the stable arm converging to it from north-east whenever
the initial values (µ (t) , χ (t))|t=0 are suitably chosen. A detailed discussion on the complete
phase diagram, including the other two non attractive steady states for system (32), under the assumption
θα (µs)α−1 < ρ < θα (µ∗
)α−1, (35)
where µs will be defined in (42), can be found in [19]; Figure 1 on p. 266 there illustrates such
phase diagram (see also Figure 2 below).
According to [17], to solve (32) we eliminate time and tackle the ODE given by the ratio between the equations in (32):
χ′
(µ) = [(αθµ
α−1− ρ) /σ] Q (µ) − 2αβ (1 − α) [θµα− χ (µ)]
[Q (µ) − 2β (1 − α) µ] [θµα− χ (µ)] χ (µ) , (36)
where Q (µ) is defined in (33). Following [20], the solution of (36), yielding the optimal policy function ˜χ (µ) along the turnpike, is approximated through a projection method based on OLS applied to the integral of a residual function built upon an approximation function which is a linear combination of n Chebyshev polynomials.9 Therefore, our estimate of the policy ˜χ (µ)
turns out to be a polynomial of degree n.
9The integral of the residual function is itself approximated by means of Gauss-Chebyshev quadrature on
the relevant interval, while as initial condition for the Maple 16 nonlinear programming (NLP) solver with the sequential quadratic programming (sqp) method we use a Chebyshev regression of order n (Algorithm 6.2 on p. 223 in [16]) on the line crossing the two steady states (µ∗, χ∗) and (µs, χs), with coordinates defined in (34)
Using (30) and (31), the optimal consumption policy for problem (27) corresponding to ˜χ (µ) is thus obtained as ˜ c (A) = ˜χ (µ) A = ˜χ α β (1 − α) 1 + 2 2A − 3 A. (37)
To approximate the time-path trajectory ˜µ (t), ˜χ (µ) is substituted into the first equation of (32) so to obtain a ODE with respect to time which can be numerically solved through the standard Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant method available in Maple 16. The time-path trajectory ˜χ (t) is then computed as ˜χ (t) = ˜χ [˜µ (t)], while the time-path trajectories of the stock of knowledge and capital, according to (30) and (20), are given by
˜
A (t) = α/ [β (1 − α) ˜µ (t) − α]+3/2 and ˜k (t) = ˜kh ˜A (t)i respectively. Similarly, the time-path trajectory of output is given by ˜y (t) = θh˜k (t)iαh ˜A (t)i1−α, while, using (31), the time-path trajectory of the optimal consumption is obtained as ˜c (t) = ˜χ (t) ˜A (t). Finally, according to (23), the time-path trajectory of optimal investment into new knowledge production is given by ˜J (t) = [˜y (t) − ˜c(t)] ϕh ˜A (t)i/n˜k′h ˜
A (t)i+ ϕh ˜A (t)io.
4.2
Optimal Dynamics Toward Stagnation
Trajectories initiating on a point A0, ˜k (A0)
on the turnpike and evolving toward the steady state A0, ˆk (A0)
on the stagnation line according to a constant zero-R&D investment policy, J (t) ≡ 0 [corresponding to condition δ/ϕ (A) < λ in (19)], are just standard saddle-path stable trajectories of a typical Ramsey model for a given (constant) level A0 of knowledge stock. That
is, they are solutions of the following social planner problem in the two variables k (state) and c (control), and the usual dynamic constraint:
¯ V (A0) = max [c] Z ∞ 0 e−ρtc1−σ − 1 1 − σ dt (38) subject to ˙k = θkα A1−α0 − c, k (0) = ˜k (A0) ,
in which the initial condition is the capital value on the turnpike corresponding to A0, k (0) =
˜
k (A0). Although, as occurs in (27), the value function ¯V in (38) depends only on the initial
stock of knowledge A0, the true initial condition of the Ramsey problem here is the initial stock
of capital ˜k (A0) – itself a function of A0 – because, unlike (27), the only state variable in (38)
is physical capital, k, while the stock of knowledge remains constant at level A0.
We actually solve a problem which is equivalent to (38) but have variables rescaled by the ratios µ = k/A0 and χ = c/A0, with A0 constant. This choice allows for studying the phase
diagram of the dynamics of (38) in the same ‘detrended’ (µ, χ) space that contains the optimal policy ˜χ (µ) of model (27) previously built. A constant stock of knowledge A ≡ A0 implies
˙
A ≡ 0, which, in turn, allows to suitably rewrite the necessary conditions for the current-value Hamiltonian associated to (38) in terms of µ = k/A0 and χ = c/A0 according to the following
system of ODEs describing the detrended optimal dynamics:
˙µ = θµα− χ
Again we eliminate time by taking their ratio and study the unique ODE characterizing the optimal policy, ¯χ (µ), in this scenario:
χ′
(µ) = χ (µ) θαµα−1− ρ / {σ [θµα
− χ (µ)]} . (40) To approximate the solution of (40) we apply a projection method based on Chebyshev Or-thogonal Collocation on n collocation points applied to a residual function built upon an ap-proximation function which is a linear combination of n Chebyshev polynomials.10 Thus, also
here our estimate of the policy ¯χ (µ) turns out to be a polynomial of degree n, possibly with a different n than that used in the previous Subsection. The optimal consumption policy for problem (38), which is a function of the state variable k, is then obtained as
¯
c (A0, k) = ¯χ (k/A0) A0, (41)
where also its dependency on the initial stock of knowledge A0 has been emphasized.
4.3
A Comprehensive Detrended Phase Diagram
Figure 2 reports a unique phase diagram in the detrended (µ, χ) space exhibiting all loci in-volved,11 the three relevant steady states, and both optimal policy curves, ˜χ (µ) along the
turnpike and ¯χ (µ) toward stagnation (the thick curves in black and dark grey respectively), for the parameters’ values considered in Section 7. The saddle-path stable steady state (µ∗, χ∗)
of the ˜χ (µ) policy with coordinates given by (34) lies on the bottom left, the steady state12
(µs, χs) of the ˜χ (µ) policy with coordinates
µs=1 + 2α +√1 + 4α + α2/ [3β (1 − α)] and χs = θ (µs)α, (42)
lies on the top right, while between these two there is the steady state (ˆµ, ˆχ) with coordinates ˆ
µ = (θα/ρ)1/(1−α) and χ = θ (θα/ρ)ˆ α/(1−α), (43) which happens to be irrelevant for the ˜χ (µ) policy but turns out to be the unique saddle-path stable steady state for the ¯χ (µ) policy defining the optimal path (the stable arm) for system (39). The latter steady state, (ˆµ, ˆχ), with coordinates in (43) corresponds to any steady state A, ˆk (A) on the stagnation line defined in (22) to which the economy might eventually
10Details can be found in Chapter 11 in [16], in Chapter 6 in [12], or in Paragraph 5.5.2 in [18]. As initial
condition for the Maple 16 ‘fsolve’ routine used to numerically solve the system of n + 1 equations setting the residual function equal to zero on each collocation node plus the steady state constraint, ¯χ (ˆµ) = ˆχ, here we use a Chebyshev regression of order n (Algorithm 6.2 on p. 223 in [16]) on the line tangent to the optimal policy
¯
χ (µ) on the steady state (ˆµ, ˆχ) with coordinates defined in (43). The slope of ¯χ (µ) on the steady state (ˆµ, ˆχ) is the positive solution of the quadratic equation obtained through l’Hˆopital’s rule, according to [4], pp. 595–596.
11All loci are plotted as thin black curves. Both the vertical line at µ∗, with µ∗ defined in (34), and the flat
increasing curve, defined as χ = θµα, crossing points (ˆµ, ˆχ) and (µs, χs) determine the ˙µ = 0 locus for system
(32). The curve with more pronounced concavity crossing all three steady states is the unique ˙χ = 0 locus for system (32), defined by (60) in [19]. Note that the curve χ = θµα defines the ˙µ = 0 locus for system (39) as
well. Finally, the vertical line at ˆµ, with ˆµ defined in (43), is the ˙χ = 0 locus for system (39).
12The steady state (µs, χs), with coordinates defined in (42), cannot be classified analytically as the Jacobian
matrix of (32) evaluated at (µs, χs) has some elements that diverge either to −∞ or to +∞, the sign of infinity
depending on the direction along which (µs, χs) is approached. For this reason, such point has been called
‘supersingular’ by Privileggi [19]. As a matter of fact, it turns out to be harmless, as the optimal policy ˜χ (µ) simply crosses it. See Remark 1 on p. 266 and the discussion on p. 267 in [19] for a more thorough description.
converge; that is, it is the unique representation in the (µ, χ) space of all steady statesA, ˆk (A) – i.e., all points on the stagnation line ˆk (A) – for the optimal dynamics of problem (38) in the (A, k) space.13 µ χ µ∗ ˆ µ µs µ ℓ χ∗ ˆ χ χs ˙µ = 0 ˙µ = 0 ˙χ = 0 ˙χ = 0 ˜ χ (µ) ¯ χ (µ)
Figure 2: phase diagram for the policy along the turnpike, ˜χ (µ) (thick black curve), and for the policy toward stagnation, ¯χ (µ) (thick dark grey curve), including all loci and steady states, for the
parameters’ values used in Section 7.
While Figure 2 shows the whole optimal policy ˜χ (µ) (the black thick curve) starting on any point on the turnpike and evolving along it to eventually converge to its steady state (µ∗
, χ∗
), only the upper right branch of the saddle-path stable arm crossing the steady state (ˆµ, ˆχ) of the optimal dynamics defined by (39) (the dark grey thick curve) is reported. That is, only paths starting from initial values µ0 > ˆµ and χ0 > ˆχ are considered here for the optimal policy
¯
χ (µ). This is because, from Proposition 2, the relevant range for A0 is the interval
3/2, ˆA, where ˆA is the knowledge value at which the turnpike ˜k (A) intersects the stagnation line ˆk (A) from above, as calculated in (26); therefore, ˜k (A) > ˆk (A) for all 3/2 < A < ˆA, so that we are considering only monotonically decreasing time-path capital trajectories, k (t), converging from above to the steady state A0, ˆk (A0)
, all corresponding to the unique optimal policy ¯χ (µ) starting from any value (µ0, χ0) to the north-east of the unique steady state (ˆµ, ˆχ) at t = 0
and then converging to it through a decreasing pattern of both time-path trajectories ¯µ (t) and ¯
χ (t).14
13From (22) and the definition µ = k/A it is clear that any steady capital value on ˆk (A) in the (A, k) space
must correspond to a point on the vertical line at ˆµ in the (µ, χ) space, with ˆµ defined in (43). On the other hand, it is immediately seen from the system of ODEs describing the optimal dynamics for problem (38) in the (k, c) space – corresponding to system (39) in the (µ, χ) space – that all steady consumption values on the stagnation line are given by ˆc (A) = θ (θα/ρ)α/(1−α)A. Hence, under the transformation χ = c/A, all such steady consumption values correspond to the unique ˆχ value in the (µ, χ) space defined in (43).
14Note that all possible optimal policies for problem (38) – each depending on a different initial condition A 0
and belonging to a different phase diagram in the (k, c) space – after being transformed into the dynamics defined by system (39) have the unique phase diagram representation in the (µ, χ) space as in Figure 2, portraying the
4.4
Welfare Matching
To find the point Am corresponding to the unique Skiba-point lying on the turnpike – that is,
the knowledge level satisfying ksk(A
m) = ˜k (Am) – we must solve the equation ˜V (A) = ¯V (A),
where ˜V and ¯V are the value functions defined in (27) and (38) respectively.
To approximate the latter, note that, for any given (constant) A, (38) is a standard Ramsey model in the state variable k and control variable c with initial condition k0 = ˜k (A). Hence,
under Assumptions A.2 and A.3 it is a concave problem, so that the assumptions in Theorem 2 of Section 2 are satisfied and we can compute the derivative of the value function at the initial stock of capital value through (5):
¯ V′ (k0) = ¯V′h˜k (A)i = u′ θh˜k (A)iαA1−α− · ¯ k0+; ˜k (A) =h¯cA, ˜k (A)i −σ , (44) where · ¯
k0+; ˜k (A)denotes the optimal initial investment toward stagnation and ¯cA, ˜k (A)
is the estimated value of the optimal policy at (A, k0) =
A, ˜k (A) numerically obtained in (41). By replacing (44) into the Hamilton-Jacobi-Bellman equation (6) of Corollary 1, we obtain the value function of problem (38) directly as a function of A:
¯
V (A) = (1/ρ)u [¯c(A, k0)] + ¯V′(k0)θk0αA1−α− ¯c(A, k0)
= 1 ρ h ¯ cA, ˜k (A)i1−σ − 1 1 − σ + θh˜k (A)iαA1−α− ¯cA, ˜k (A) h ¯ cA, ˜k (A)iσ . (45)
Because problem (27) turns out to be not concave in early-time dynamics, Theorem 2 is not directly applicable to approximate the value ˜V (A). Therefore, we rely on an ad-hoc approach based on guessing a candidate value function and then checking that it satisfies the assumptions of Theorem 1 in Section 2. As Theorem 2 provides only sufficient conditions for the differentiability of the value function, we are allowed to build our guess candidate by assuming that it is differentiable with derivative given by (5), and then again define ˜V (A) according to the the Hamilton-Jacobi-Bellman equation (6), as we did for ¯V (A) in (45). Hence, we first set
w′ (A) = u′ θh˜k (A)iαA1−α−h˜k′ (A) + ϕ (A)i · ˜ A 0+; A h˜k′ (A) + ϕ (A)i = k˜ ′ (A) + ϕ (A) [˜c (A)]σ , (46) where · ˜
A (0+; A) denotes the optimal knowledge change on the turnpike at A according to (28),
and ˜c (A) is the estimated value of the optimal policy along the turnpike at A numerically obtained in (37). Next, we use (28) to replace (46) into (6) and obtain our candidate guess for
same qualitative properties of all different phase diagrams in the (k, c) space, each characterized by a different steady state, stable arm, etc., depending on A0.
the value function of problem (27): w (A) = 1 ρ u [˜c (A)] + w′ (A) θh˜k (A)iαA1−α− ˜c(A) ˜ k′(A) + ϕ (A) = 1 ρ [˜c (A)]1−σ − 1 1 − σ + θh˜k (A)iαA1−α− ˜c(A) [˜c (A)]σ . (47)
Note that (47) yields the same expression of (45), only with the optimal policy value along the turnpike, ˜c (A), in place of the optimal policy value toward stagnation, ¯cA, ˜k (A).
Proposition 4 Whenever the parameters’ values in Assumptions A.1–A.3 satisfy
θα [β (1 − α) /α]1−α < (1 + σ) ρ, (48) w (A) as defined in (47) is the value function ˜V (A) of problem (27).
Proof. See the Appendix.
Remark 2 Under condition (48) of Proposition 4, using (47) and (45) in order to estimate welfare for both the model converging to the ABGP and the model converging toward stagnation require only one numerical step for each model: the projection method to approximate the optimal policies according to (37) and (41) as discussed in Subsections 4.1 and 4.2. Specifically, in the following Algorithm 1 no time-path trajectories estimations are needed.
Remark 3 Proposition 4 indirectly establishes that when (48) holds the value function of prob-lem (27) is differentiable although assumption (ii) of Theorem 2 does not hold, as the instan-taneous felicity U (·, ·) turns out to be not concave for small values of A; specifically, for those contained in the range 3/2, ˆAi. Figure 3 below shows that the value function ˜V (A) = w (A) itself turns out to be convex over 3/2, ˆAi, while it becomes concave for larger values of A.
4.5
Skiba-Point Estimation
Assume that condition (48) holds and let
f (A) = w (A) − ¯V (A) , (49) with w (A) and ¯V (A) defined in (47) and (45) respectively, be the function whose unique zero is to be found. In order to bracket this zero we must take an initial interval large enough to contain it; such a choice requires to find a left endpoint value Aℓ > 3/2 sufficiently close to15
3/2 so that f (Aℓ) < 0, while as right endpoint a good value is given by ˆA in (26), because
Proposition 2 implies that f ˆA> 0. To Aℓ corresponds the right endpoint µℓ = ˜k (Aℓ) /Aℓ of
the range for variable µ in the phase diagram of Figure 2. All steps to numerically approximate the unique Skiba-point on the turnpike are summarized in Algorithm 1 below.
15It has been observed in [20] that the ˜χ (µ) approximation becomes less reliable for larger µ values; thus, the
choice of Aℓ, should not be too small so to keep the error in the projection method described in Subsection 4.1
Algorithm 1 (Finds Am satisfying ksk(Am) = ˜k (Am))
Step 1: Choose a value Aℓ > 3/2 sufficiently close to 3/2; the range for the ˜χ (µ) policy
approximation in the (µ, χ) space is [µ∗, µ
ℓ], with µ∗ as in (34) and, according to (30),
µℓ = {α/ [β (1 − α)]} [1 + 2/ (2Aℓ− 3)]. The range for the ¯χ (µ) policy approximation is
[ˆµ, µℓ], with ˆµ defined in (43), corresponding to
h Aℓ, ˆA
i
in the (A, k) space.
Step 2: Apply the OLS-Projection method discussed in Subsection 4.1 to estimate the optimal policy along the turnpike, ˜χ (µ), on the range [µ∗
, µℓ].
Step 3: Apply the Collocation-Projection method discussed in Subsection 4.2 to estimate the optimal policy toward stagnation, ¯χ (µ), on the range [ˆµ, µℓ].
Step 4: Use policies ˜χ (µ) and ¯χ (µ) evaluated in steps 2 and 3 to compute ˜c (A) as in (37) and ¯
cA, ˜k (A)as in (41) so to get w (A) and ¯V (A) according to (47) and (45) respectively; define f (A) as in (49).
Step 5: Apply the standard Maple 16 ‘fsolve’ routine to equation (49) using the range hAℓ, ˆA
i
to find Am satisfying f (Am) = 0.
Step 6: Report the solution, Am, and evaluate the Skiba-point, ksk(Am) = ˜k (Am).
The Maple 16 code for Algorithm 1 is available from the author upon request.
Figure 3(a) plots both value functions ˜V (A) = w (A) (in black) and ¯V (A) (in dark grey) of problems (27) and (38) respectively as approximated through Algorithm 1 on the range h
Aℓ, ˆA
i
for the parameters’ values considered in Section 7. As problem (27) is not concave for small values of the stock of knowledge A, consistently, ˜V (A) turns out to be convex on such initial values range. Do not be misled by the convexity of the value function ¯V (A) of the model leading to stagnation: problem (38) is a standard concave Ramsey problem in its state variable, which is physical capital, k; in fact, its value function as a function of k is definitely concave. Function ¯V (A), as a function of A, represents a whole family of Ramsey problems, each indexed by the value A (initial stock of knowledge) which remains constant as physical capital, k, evolves through time toward its steady value ˆk (A). It turns out that the value functions of these problems evolve in a convex fashion as A increases in the range hAℓ, ˆA
i . Figure 3(b) plots the same value functions for a range of A-values larger than hAℓ, ˆA
i ; they are obtained through Algorithm 1 where a larger interval µ, µℓ, with µ < ˆµ, has been chosen
in step 3. It is clearly seen that ˜V (A) becomes concave as A increases, that is, when the unit cost of knowledge production ϕ (A) in (12) approaches its asymptotic constant value 1/β, or, equivalently, when the turnpike ˜k (A) approaches the asymptotic turnpike ˜k∞(A).
5
Skiba-Points Above the Turnpike
According to Proposition 3, to the left of the knowledge level Am found in the previous section
the Skiba-point necessarily must lie strictly ‘above’ the turnpike, i.e., ksk(A0) > ˜k (A0) for all
Aℓ ≤ A0 < Am. Thus, for values of the initial stock of knowledge A0 < Amwe must characterize
A Aℓ Am Aˆ ˜ V (A) ¯ V (A) (a) A Am ˜ V (A) ¯ V (A) (b)
Figure 3: value functions ˜V (A) (in black) and ¯V (A) (in dark grey) of problems (27) and (38) for the parameters’ values used in Section 7, (a) on the range Aℓ, ˆA
i
and (b) for larger values of A.
later instant t0 > 0, after which the economy continues its evolution according to optimal
trajectories of the sort discussed in Subsection 4.1.
Any optimal trajectory above the turnpike must satisfy the last necessary condition in (19), δ/ϕ (A) > λ, corresponding to the largest possible investment in R&D activities by the social planner:16 J = y = θkαA1−α. In other words, along such early-transition trajectories it
is optimal to invest all the output into the production of new knowledge. Hence, on the time interval [0, t0] problem (17) simplifies to one in two state, A and k, and one control, c, variables:
max [c] Z t0 0 e−ρtc1−σ− 1 1 − σ dt (50) subject to ˙ A = θkαA1−α/ϕ (A) ˙k = −c A (t0) = Ar, k (t0) = ˜k (Ar) , c (t0) = ˜c (Ar) ,
with the additional constraint 0 ≤ c ≤ k, where again the time argument has been dropped for simplicity, Ar > A0 is the knowledge level corresponding to instant t0 > 0 at which the
turnpike is hit from above, ˜k (Ar) is the corresponding capital value on the turnpike and ˜c (Ar)
is the optimal policy value for consumption on the turnpike at Ar according to (37). Instead of
initial conditions, three terminal conditions are given for problem (50) that bound the optimal trajectories to land on the turnpike at the point Ar, ˜k (Ar)
at instant t0. While the first two
are obvious, the last one, c ˜t = ˜c(Ar), stating that the terminal value of consumption must
match the optimal consumption value on the turnpike, holds because the control c of problem (17) must be continuous for all t ≥ 0, as noted in Remark 1.
The Skiba-point corresponding to some initial stock of knowledge Aℓ ≤ A0 < Am, is the
initial capital value k0 = ksk(A0) that equates the welfare produced by the whole optimal
consumption time-path trajectory, for t ∈ [0, +∞), that starts on (A0, k0) at t = 0 and it is
the piecewise union of the optimal early transition trajectory above the turnpike over [0, t0]
with the optimal transition trajectory along the turnpike over (t0, +∞), with the welfare
gen-erated, according to δ/ϕ (A) < λ in (19), by a constant zero-R&D investment policy, J ≡ 0, starting from the same initial point (A0, k0) and leading the economy toward the steady state
A0, ˆk (A0)
on the stagnation line. Clearly, the former trajectory is defined by the intersection
value Ar at the positive instant t0. Thus, our aim is to build an iterative algorithm to estimate
Ar and the corresponding positive instant t0 > 0 that determines an optimal whole time-path
trajectory toward steady growth starting at (A0, k0) yielding the same welfare of that
start-ing as well from (A0, k0) but leading to stagnation. In other words, we start by an arbitrary
choice of Ar and study the union of the optimal early transition trajectory originating from
Ar, ˜k (Ar)
at some instant t0 > 0 and, by going backward in time, defines a pair of initial
values (A0, k0) at t = 0, with its continuation along the turnpike for t > t0; next, we compare
the welfare generated by such whole trajectory with that produced by the optimal trajectory that starts from the same initial point (A0, k0) and leads to the steady state
A0, ˆk (A0)
. Substituting J with θkαA1−α in the necessary conditions for the current-value Hamiltonian
(18) we are led to the following optimal dynamics associated to (50): ˙ A = θkαA1−α/ϕ (A) ˙k = −c ˙c = cθα (k/A)α−1 − ρ /σ, (51)
which, together with the three terminal conditions, is a Cauchy problem in the three variables k, A and c. To solve system (51) again we eliminate time by taking the ratios ˙k/ ˙A and ˙c/ ˙A and study the following system of two ODEs in the functions k (A) and c (A):
k′
(A) = − c (A) ϕ (A) θ [k (A)]αA1−α
c′(A) = c (A) ϕ (A)θα [k (A) /A] α−1
− ρ σθ [k (A)]αA1−α .
(52)
To solve (52) we first choose the initial stock of knowledge Aℓ < A0 < Am– with Aℓbeing the
lower bound used in Algorithm 1 and Am the estimate generated by the same Algorithm – and a
value Ar> A0. We then apply a projection method based on Chebyshev Orthogonal Collocation
on n collocation points over the interval [A0, Ar] applied to the two residual functions – one
for each policy kab(A) and cab(A) to be estimated – built upon approximation functions which
are linear combinations of n Chebyshev polynomials.17 Thus, our estimates of the two policies
kab(A) and cab(A) are polynomials of degree n, possibly with a different n than those used
in the previous subsections. To approximate the optimal time-path trajectory of the stock of knowledge Aab(t) along this early transition dynamic for the economy, kab(A) is substituted
into the first equation of (51) so to obtain a ODE with respect to time which can be numerically solved through the standard Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant method available in Maple 16. The corresponding optimal time-path trajectories kab(t), yab(t), Jab(t) and cab(t) are then computed as kab(t) = kabAab(t), yab(t) = Jab(t) =
θkab(t)α
Aab(t)1−α
and cab(t) = cab[A (t)] respectively. Finally, the instant t
0 at which the
optimal trajectories just evaluated hit the turnpike on the point Ar, ˜k (Ar)
is approximated by solving Aab(t) = A
r with respect to t through the Maple 16 ‘fsolve’ routine. The initial
17As initial condition for the Maple 16 ‘fsolve’ routine used to numerically solve the system of 2n+2 equations
setting the two residual functions equal to zero on each collocation node plus the two terminal conditions kab(A
r) = ˜k (Ar) and cab(Ar) = ˜c (Ar), we use a Chebyshev regression of order n (Algorithm 6.2 on p. 223 in
[16]) on the lines crossing the pairs of points A0, ˜k (A0)
, Ar, ˜k (Ar)
and (A0, ˜c (A0)), (Ar, ˜c (Ar)) for the
kab(A
capital level corresponding to A0 at t = 0 along the backward-in-time trajectory starting from Ar, ˜k (Ar) is thus computed as k0(A0, Ar) = kab(A0) , (53)
where kab(A) is the capital optimal policy solving (52).
The whole optimal transition time-path trajectories ˜Aab(t), ˜kab(t), ˜yab(t), ˜Jab(t) and ˜cab(t)
for all t ≥ 0 when the economy starts at t = 0 from the initial conditions (A0, k0(A0, Ar)),
with k0(A0, Ar) as in (53), can be built as piecewise functions by joining each trajectory above
the turnpike over [0, t0] with its ‘continuation’ along the turnpike over (t0, +∞) at the instant
t = t0, ˜ zab(t) = z ab(t) for t ∈ [0, t 0] ˜ z (t) for t ∈ (t0, +∞) , (54)
with zab ∈ Aab, kab, yab, cab, Jab , while all the ˜z ∈ n ˜A, ˜k, ˜y, ˜c, ˜Jo time-path trajectories are
built according to the method discussed at the end of Subsection 4.1 on the range [µ∗
, µr], with
µ∗ as in (34) and µ
r = ˜k (Ar) /Ar. As ˜Jab(t) = yab(t) for t ∈ [0, t0] while ˜Jab(t) = ˜J (t) < ˜y (t)
for t ∈ (t0, +∞), with ˜J (t) given by (23), we expect to observe a discontinuity ‘jump’ for
the optimal control ˜Jab at instant t
0, as postulated by necessary conditions (19), while all
other trajectories must exhibit a kink on t0, where they are not differentiable. This pattern is
confirmed in Figure 6 of Section 7.
To estimate welfare when the economy follows its path along the turnpike toward the ABGP starting at t = 0 from (A0, k0(A0, Ar)), with k0(A0, Ar) defined by (53), we apply Lemma 1 of
Section 2 and again Proposition 4. Specifically, under condition (48) we conveniently split it as the sum of two terms:
˜ Vab(A0, Ar) = Z t0 0 e−ρtc ab(t)1−σ − 1 1 − σ dt + e −ρt0 w (Ar) , (55)
where w (Ar) = ˜V (Ar) is the value function of problem (27) according to (47) of Subsection 4.4
evaluated at the intersection point Ar. That is, at t = t0 we consider the welfare generated by
the economy along the turnpike when it starts with initial stock of knowledge Ar, and discount
this value in t = 0. The first integral on the RHS of (55) is approximated through a Gauss-Legendre quadrature routine on a large number of nodes over the time range [0, t0], using the
time-path trajectory value of optimal consumption, cab(t), defined before on each node.
To calculate welfare when the economy converges to the steady state A0, ˆk (A0)
on the stagnation line when starting at t = 0 from the same initial point (A0, k0(A0, Ar)), with
k0(A0, Ar) defined by (53), we restate problem (38) according to
¯ Vab(A0, Ar) = max [c] Z ∞ 0 e−ρtc1−σ− 1 1 − σ dt (56) subject to ˙k = θA1−α 0 kα− c, k (0) = k0(A0, Ar) .
Because the ‘detrended’ system in the ratio variables µ = k/A0 and χ = c/A0 associated
to the optimal dynamics of (56) turns out to be the same as in (39), the optimal policy for (56) is obtained according to the same approximation procedure discussed in Subsection 4.2 by means of (41), that is, ¯c (A0, k) = ¯χ (k/A0) A0, using k (0) = k0(A0, Ar), with k0(A0, Ar)
starting from initial capital values above the turnpike, k0(A0, Ar) > ˜k (A0), the
Collocation-Projection method must be performed over a range [ˆµ, µ0] larger than the interval [ˆµ, µℓ] used
in Subsection 4.5. Specifically, when the initial stock of knowledge equates the lower bound A0 = Aℓ used in Algorithm 1, setting µ0 = k0(Aℓ, Ar) /Aℓ implies that µ0 > µℓ = ˜k (Aℓ) /Aℓ
whenever k0(A0, Ar) > ˜k (Aℓ), as will be the case in one of our simulations of Section 7.
To approximate ¯Vab(A
0, Ar) in (56) we apply the same technique explained in Subsection
4.4, based on Theorem 2 and Corollary 1, and again exploit the Hamilton-Jacobi-Bellman equation: ¯ Vab(A0, Ar) = {¯c[A0 , k0(A0, Ar)]}1−σ− 1 ρ (1 − σ) + θ [k0(A0, Ar)]αA01−α− ¯c[A0, k0(A0, Ar)] ρ {¯c[A0, k0(A0, Ar)]}σ , (57) where ¯c [A0, k0(A0, Ar)] is the estimated value of the optimal consumption policy at the initial
point (A0, k0(A0, Ar)) numerically obtained by (41).
Algorithm 2 below summarizes all the steps discussed above. Because a complex pointwise estimation for each welfare value ˜Vab(A
0, Ar) required by the integral approximation in (55), it
relies on a standard Bisection Method (see, e.g., Algorithm 5.1 on p. 148 in [16]). Fix a given initial stock of knowledge Aℓ ≤ A0 < Am and let
fab(A
r) = ˜Vab(A0, Ar) − ¯Vab(A0, Ar) , (58)
with ˜Vab(A
0, Ar) and ¯Vab(A0, Ar) defined in (55) and (57) respectively, be the function whose
unique zero is the target of our search routine. For each given A0, the unique value A∗r such
that fab(A∗
r) = 0 yields our estimation of the Skiba-point as
ksk(A0) = k0(A0, A∗r) , (59)
where k0(A0, Ar) is defined according to (53). As ksk(A0) > ˜k (A0), fab(A0) < 0 must hold;
hence, A0 plays a useful role as left endpoint of the initial interval bracketing the unique zero
of fab. The following result is useful to estimate the right endpoint of such interval.
Proposition 5 For any initial stock of knowledge A0 such that Aℓ ≤ A0 < Am, the unique A∗r
such that fab(A∗
r) = 0 must satisfy A ∗
r ≥ Am.
Proof. Suppose, on the contrary, that A∗
r < Am. Let ksk(A0) as in (59) be the Skiba-point
associated to the initial knowledge level A0, then the trajectory starting at t = 0 on the initial
point A0, ksk(A0), with A0 < Am, and hitting the turnpike at a later instant t0 > 0 on ˜k (A∗r)
yields the same welfare as the trajectory leading to stagnation from A0, ksk(A0). In Section
4 Am has been defined as the unique value satisfying ksk(Am) = ˜k (Am); therefore, according to
Proposition 3, A∗
r < Am implies that the trajectory converging to the steady state
A∗ r, ˆk (A ∗ r) on the stagnation line according to a zero-R&D investment, J ≡ 0, policy for t > t0 yields a
larger welfare than the trajectory continuing along the turnpike toward steady growth. This contradicts the assumption that ksk(A
0) is the Skiba-point.
In view of Proposition 5, before starting the true Bisection Method we will perform a number of preliminary iterations to estimate the right endpoint of the interval bracketing the zero of fab, starting from A
R = Am and then increasing this value by a (small) constant increment
Algorithm 2 (Finds the Skiba-point when A0 < Am)
Step 1: Set the range [µ∗
, µℓ] as in step 1 of Algorithm 1 for the ˜χ (µ) policy approximation
in the (µ, χ) space. The range for the ¯χ (µ) policy approximation is [ˆµ, µ0], with ˆµ as in
(43) and µ0 = µℓ+ ϑ, with ϑ sufficiently large to allow the estimation of ¯c [A0, k0(A0, Ar)]
according to (53) in the following step 4.2.7 when k0(A0, Ar) > ˜k (Aℓ); i.e., µ0 must
satisfy k0(A0, AR) ≤ µ0A0.
Step 2: Apply the OLS-Projection method discussed in Subsection 4.1 to estimate the optimal policy along the turnpike, ˜χ (µ), on the range [µ∗
, µℓ].
Step 3: Apply the Collocation-Projection method discussed in Subsection 4.2 to estimate the optimal policy toward stagnation, ¯χ (µ), on the range [ˆµ, µ0].
Step 4: Find Ar satisfying fab(Ar) = 0 for fab defined in (58).
Step 4.1 (Initialization): Set [AL, AR] = [A0, Am], with Am being the output of
Algo-rithm 1, as the initial interval for searching the interval bracketing the zero of fab
in (58), set a (switch) variable B = 1, choose tmax > 0 for the range of the
Runge-Kutta routine in the following step 4.2.3, choose N > 0 as the number of nodes for the Gauss-Legendre quadrature routine in the following step 4.2.5, choose an incre-ment ǫ > 0, choose stopping rule parameters 0 < ε, η < 1, and set (fake) initial values fab(A
r) = fab(AR) = 1 > η.
Step 4.2 (Bisection loop): While AR− AL> ε and
fab(Am)
> η do:
1. if B = 1 then set AR = AR+ ǫ (increase right bound) and Ar = AR, else set
Ar= (AR− AL) /2 (compute midpoint),
2. approximate policies kab(A) and cab(A) over [A
0,Ar] by solving (52) through the
Collocation-Projection method described above,
3. use kab(A) and cab(A) from step 4.2.2 to build the time-path trajectories Aab(t)
and cab(t) over [0, t
max] through the Runge-Kutta routine as explained above,
4. find t0 by solving Aab(t) = Ar through Maple 16 ‘fsolve’ routine over [0, tmax],
5. apply the Gauss-Legendre quadrature routine explained before to approximate the integral in (55), use ˜χ (µ) from step 2 to evaluate ˜c (A) through (37), compute w (Ar) according to (47) and evaluate ˜Vab(A0, Ar) as in (55),
6. evaluate k0(A0, Ar) using kab(A) from step 4.2.2 to according to (53),
7. use ¯χ (µ) from step 3 to evaluate ¯c [A0, k0(A0, Ar)] through (41) and evaluate
¯ Vab(A 0, Ar) by means of (57), 8. update fab(A r) by setting fab(Ar) = ˜Vab(A0, Ar) − ¯Vab(A0, Ar), 9. if B = 1 and fab(A
r) < 0 then (keep searching for bracket right endpoint) go to
step 4.2, else (bisection loop)
- if B = 1 set B = 0 (stop searching for bracket), - refine the bounds: if fab(A
r) fab(AR) < 0 then set AL = Ar, else set AR= Ar
and update fab(A
R) by setting fab(AR) = fab(Ar).
Step 5: Report the Skiba-point from step 4.2.6, ksk(A
Remark 4
1. The choice of tmax in step 4.1 is a delicate issue, because it depends on the range [A0, Ar]
over which the Projection Method approximates the policies kab(A) and cab(A) in step
4.2.2. If it is too large, the Runge-Kutta algorithm in step 4.2.3 stops too early yielding an error message because it tries to estimate a trajectory continuing beyond the intersection point Ar, ˜k (Ar)
on the turnpike, which after a short while ceases to be defined. On the other hand, if it is too small it fails to catch the t0 value, which, indeed, happens to be
close to tmax. Hence a suitable tmax value should be chosen through some guess-and-tries.
2. The degree of approximation, n, in the Collocation-Projection method performed in step 4.2.2 must be smaller for A0 values closer to Am, as too many Chebyshev polynomials in
a small interval cause the algorithm to stall.
The Maple 16 code for Algorithm 2 is available from the author upon request.
6
Skiba-Points Below the Turnpike
For values of initial stock of knowledge to the right of Am Proposition 3 implies that the
Skiba-point must lie strictly ‘below’ the turnpike, i.e., ksk(A
0) < ˜k (A0) for all Am < A0 < ˆA, where
Am is given by Algorithm 1 and ˆA is defined in (26). This type of scenario forecasts optimal
early transition trajectories starting below the turnpike at t = 0 and entering the turnpike from below at some later instant t0 > 0, after which the economy continues its evolution along the
turnpike according to optimal trajectories of the sort discussed in Subsection 4.1. The former trajectories are characterized by δ/ϕ (A) < λ in (19) and thus envisage a zero investment policy in new knowledge production: J (t) ≡ 0 and A (t) ≡ A0 for all t ∈ [0, t0]. In other words, like
along trajectories converging to the point A0, ˆk (A0)
on the stagnation line, the economy evolves through time along the vertical line A ≡ A0 in the (A, k) space, though in the opposite
direction (i.e., moving upward), accumulating physical capital until the turnpike is reached. Because A remains constant at the A0level when t ∈ [0, t0], these optimal dynamics restated
in terms of ratio variables µ = k/A and χ = c/A must satisfy the associated short-run necessary conditions described by (39) in the (µ, χ) space, so that they can be represented in the same phase diagram of the ˜χ (µ) and ¯χ (µ) policies discussed in Subsections 4.1 and 4.2 respectively. Specifically, setting
˜
µ0 = ˜k (A0) /A0, (60)
as the µ value corresponding to the point at which our trajectory hits the turnpike from below at t = t0, the terminal condition
χbe(˜µ0) = ˜χ (˜µ0) (61)
establishes a well defined Cauchy problem for the single ODE (40). According to Remark 1, terminal condition (61) is justified by the continuity of the optimal control χbe = c/A at the
intersection point with the turnpike.
After fixing an initial stock of knowledge A0 such that Am < A0 < ˆA we solve this problem
by means of a projection method based on OLS applied to the integral of a residual function built upon an approximation function which is a linear combination of n Chebyshev polynomials.18
18As usual, the integral of the residual function is itself approximated by means of Gauss-Chebyshev quadrature
